A wire has a length of 20 m. It will be used to make a chicken coop. For the area to be maximum, the length and width of the wire must be ... respectively.
Given the total length of the wire is 20 m. This length represents the perimeter of the rectangular chicken coop. Let the length be p and the width be l, we can write the perimeter equation:
2(p+l)=20
p+l=10⟹l=10−p
Next, we want to maximize the area of the coop (L). The area of a rectangle is L=p×l. Substitute the equation l=10−p into the area formula:
L(p)=p(10−p)
L(p)=10p−p2
This equation is a quadratic function whose curve is a downward-opening parabola (since the coefficient of p2 is negative), meaning it has a maximum value at its vertex. The coordinate of the vertex (pmax) can be found using the formula −2ab:
pmax=−2(−1)10=−−210=5 m
After finding the length p=5 m, we can determine the width l:
l=10−5=5 m
Thus, for the area to be maximum, the length and width of the coop must be 5 m and 5 m respectively.
